Optimal Control Problems For Several Classes Of Population Dynamical Systems With Size-structured Under Polluted Environment

With the rapid development of the economy,many new technologies have been developed and used in industrial and agricultural production activities,meanwhile,a large amount of toxic and harmful substances are discharged into the atmosphere,and the environmental quality is seriously affected.Environmental pollution has caused great harm to human beings,animals,plants and microorganisms.Therefore,the study of the impact of environmental pollution on biological populations has attracted the attention of many scholars.The size-structured population model can truly reflect the organic relationship between population and individual in the evolution of biological population.Forest development system is a specific size-structured population.Therefore,it has more biological significance to study them than the age-structured population model.In this paper,several kinds of models with size-structured under the influence of toxins are proposed,and the corresponding optimal control problems are analyzed.The main contents of this paper are as follows:In the first chapter,the research background and significance are described,and the research progress of population dynamics system with size-structured and the effects of toxin on population development are summarized,the preparatory knowledge and theory used in the analysis of this paper are also briefly described.The second chapter consists of four sections.The control problem of a competitive population model with size-structured under polluted environment is analyzed.Firstly,establishing model,the meaning of variables in the model is explained in detail and the corresponding assumptions are given.Secondly,the existence and uniqueness of the system solution is proved by using the characteristic method and the differential equation theory,then the existence of the optimal control is proved by using the tangent-normal cone theory.Finally,the optimality conditions of the control problem are obtained by using the Ekeland variational method and the properties of the lower semicontinuous function.In the third chapter,there are three sections.In the first section,a predator-prey model with size-structured under polluted environment is proposed.In the second section,the formal solution of the system is obtained by using the characteristic method.Then the well-posedness of the solution of the model is proved by inequality estimation and compression mapping principle.In the third section,the problem of maximum revenue control is proposed and its existence of optimal control is proved.In the fourth chapter,the optimal control problems of the linear and non-linear forest development system in the polluted environment are studied in two sections.For the linear forest development system,after the model is put forward,corresponding hypothesis and the existence and uniqueness of the solution of the model are given,then the optimal forest control problem is put forward and its existence of the optimal control is proved by the Mazur approach.As for the nonlinear forest development system,the existence and uniqueness of the model solution are analyzed by using the operator semigroup theory,and discussing the corresponding control problems by using the basic theory of the analysis.